Question: What is the remainder when $99^{36}$ is divided by 100?
Solution: Noticing that $99=100-1$ we see that \[99\equiv-1\pmod{100}.\] Therefore  \[99^{36}\equiv(-1)^{36}\equiv1\pmod{100}.\] The remainder when $99^{36}$ is divided by 100 is $\boxed{1}$.